Do Exercise 4.4.2(b). Show that the partial differential equation may be put into Sturm–Liouville form.
Reference 4.4.2(b):
In Section 4.2 it was shown that the displacement u of a nonuniform string satisfies
where Q represents the vertical component of the body force per unit length. If Q = 0, the partial differential equation is homogeneous. A slightly different homogeneous equation occurs if Q = αu.
(b) Separate variables if ρ0(x) and α(x) but T0 is constant for physical reasons.
Analyze the time-dependent ordinary differential equation.
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